Three-dimensional skyrmionic cocoons in magnetic multilayers

Three-dimensional spin textures emerge as promising quasi-particles for encoding information in future spintronic devices. The third dimension provides more malleability regarding their properties and more flexibility for potential applications. However, the stabilization and characterization of such quasi-particles in easily implementable systems remain a work in progress. Here we observe a three-dimensional magnetic texture that sits in the interior of magnetic thin films aperiodic multilayers and possesses a characteristic ellipsoidal shape. Interestingly, these objects that we call skyrmionic cocoons can coexist with more standard tubular skyrmions going through all the multilayer as evidenced by the existence of two very different contrasts in room temperature magnetic force microscopy. The presence of these novel skyrmionic textures as well as the understanding of their layer resolved chiral and topological properties have been investigated by micromagnetic simulations. Finally, we show that the skyrmionic cocoons can be electrically detected using magneto-transport measurements.

elds stronger than 200 mT and an average thickness below 2 nm, some textures that are only present in half the overall thickness (less than 7 layers) can be stabilized whereas for the uniform case, an average height below 10 layers occurs infrequently. Similar conclusions can be drawn on the intermediate case, S = 0.1 nm, for which at high eld we can reach more conned textures. To support this claim, vertical cut of the magnetization of an axisymmetric selected object are displayed below their respective graphs, for a magnetic eld of 250 mT and an average thickness close to 2 nm. We observe that for S = 0 nm, it corresponds to a columnar skyrmion (Fig. S1a) while for the other two it disappears from the outer layers and acquire a characteristic ellipsoid shape. Moreover, the range of stability in thickness seems to be broaden by the presence of the gradients as we are able to stabilize textures with lower thicknesses than the uniform case.
Finally, by introducing a variable thickness into the structure, it is possible to obtain states with isolated textures at lower eld and thickness rather than stripes or worms as shown by the circularity. Thus, overall, the use of gradient facilitates the stabilization of isolated objects which can be strongly conned under the appropriate conditions.
We also consider what we call Reversed Single Gradient (RSG) with the inverse thickness evolution, i.e S < 0. For S = −0.1 nm, the impact of the gradient seem minimal: the connement of the textures appears quite close to the uniform case. The magnetization cut shows also an object with a higher diameter 1 at equivalent eld and average thickness. A stronger eect is noticeable when using a larger step but it is restricted to a small range of thicknesses and the majority of the relaxed magnetic states corresponds to stripe states and not isolated objects, thus limiting the interest of this other possible structure. Note that we observe a magnetic object split in half in the YZ cut, with the scission happening in the layer with the stronger OOP anisotropy (of thickness 1.2 nm in that case). It shows the importance of strong PMA layers in the manipulation of the 3D magnetization distribution. However, the object is less conned than with S > 0 and the bottom and top layers display a complex behavior due to the strong in-plane eective anisotropy and the dipolar eld from the structure. Thus, based on those observations and the full numerical study, in this work we have chosen to focus on the intermediate case of SG: all results presented relate to S = 0.1 nm. Figure S1: Average height of the magnetic textures obtained in micromagnetic simulations for the a) uniform, b) SG and c) RSG structures with various S parameters at dierent magnetic elds and thicknesses with N = 13 layers. Depending on the circularity of the textures, the marker is either an open square for C < 0.9 or a circle otherwise. A zero height corresponds to a uniform magnetization. Below each graph is an associated vertical cut of the magnetization of an axisymmetric object at an external eld of 250 mT and an average thickness close to 2 nm.

Magnetic hysteresis
Field evolution The magnetic hysteresis of our samples have been measured using an alternating gradient eld magnetometer (AGFM) and displayed in Fig. S2a. As the reorientation transition thickness from out-of-plane to in-plane anisotropy is 1.7 nm in our Pt|Co|Al trilayers, the multilayers under consideration would display an in-plane eective anisotropy in a uniform state, at the exception of the PMA layers. This is conrmed by the AGFM measurements. For the SG, the easy axis lies in-plane while for the DG the situation is more complex due to the presence of the PMA layers which keep an out-of-plane magnetization Eective anisotropy To estimate the surfacial uniaxial anisotropy K u,s of our multilayers, simpler structures have been studied made of three repetitions of (Co t|Al1.4|Pt3) where t corresponds to the Cobalt thickness. From AGFM measurement, the eective anisotropy K eff was extracted which relates to K u,s with: Using the M S measured by SQUID (1.22 ± 0.02 MA/m), a linear t yields K u,s = 1.58 ± 0.08 mJ/m 2 (see  In Fig. S4b, the same zone was scanned at dierent lift heights on a DG to probe the sensitivity of the MFM with respect to the bottom layers. We nd that, for a lift height ranging from 5 to 80 nm, which corresponds roughly to the gradient thickness, the intensity of the signal for all the cocoon is typically divided by a factor 4 and their apparent size increases signicantly. Thus, the signal of the cocoons is almost negligible and all have the same behavior. This suggests that the MFM is only picking up the signal of the top layers as expected from the large thickness of our sample and the rapid decrease of the stray eld.   Between the closest states A and B, this dierence drops to 0.6 mΩ which would be harder to resolve experimentally. Therefore, the transport measurement shows some sensitivity to the 3D prole but might lack precision depending on its complexity. Similarly, in the DG, erasing the top cocoons leads to a signicant dierence in the calculated resistances, as shown in Fig. S7b. Therefore, we conclude from the match of the simulations with experiments (black points) that the cocoons must be present in both SG.

Topology and eld behavior of skyrmionic cocoons
Size evolution Skyrmionic cocoons display an important modulation over the thickness depending on the external magnetic eld. In Fig. S8, we show map of the radius r x along the x direction as a function of the eld and the magnetic layer for two dierent textures identied on the magnetization cut on the right. Their evolution is quite similar even though their initial size diers: until 100 mT they remain fairly homogeneous over the thickness but increasing the eld even more cause them to shrink signicantly in the outer layers until they disappear from them. For instance, at 275 mT for both textures, they are not present in the top three layers and they display a strong evolution over the remaining layers, giving them their typical ellipsoid shape. 2D topological properties To study the topology of the skyrmionic cocoons, a 2D topological number W in each magnetic layer is computed using a lattice-based approach [2] at dierent elds for both the SG and the DG (Fig. S9a ,b). In the continuous approach, W is dened as: In the layers in which the dipolar and DMI elds nearly cancel each other [3], W drops to zero with domain walls uniformly pointing in the same direction (see red dotted square in layer 8 in Fig. S9c). The strong PMA layers corresponds to W = −1 as a single object with Néel walls is present in the simulation space. This simple 2D topological number already hints at interesting non-trivial properties for such complex objects. 3D topological properties To further probe the topology of skyrmionic cocoons, we can consider the 3D generalization of the winding number [4,5]: Where the integral is taken over a surface S, enclosing the structure. For a single enclosed Bloch point, q takes the value ±1, and q can be zero for even number of Bloch points, like for the dipole strings, also called torons or globules. In our case, when the cocoon is fully buried into the multilayer, the magnetization on a contour is quasi uniform without no curvature, and therefore leads to q = 0 (see Fig. S10a). Alternatively, if the cocoon`emerges' at the surface of the multilayer, with a skyrmion in the topmost (or bottommost) magnetic layer, then q is entirely determined by the 2D skyrmion number of this last layer (Fig. S10b) and is thus non-zero. However, we stress that the lack of continuity of the magnetization, especially along the vertical direction, breaks the premises of the underlying mathematical foundations: in particular, q ̸ = 0 is not associated with the presence of Bloch points, which are`hidden' between the magnetic layers. As illustrated in Fig. S10c, when the symmetric exchange is low enough and the layers only coupling through dipolar exchange, there is no more continuity of the magnetization texture along the out-of-plane direction.
Therefore, the name`skyrmionic cocoons' is introduced to indicate this fundamental dierence with the continuous textures called torons. Figure S10: Calculation of the 3D topological charge, neglecting the non-continuity of the magnetization along the growth direction. a) Selection of a cubic bounding box around a cocoon fully buried in the magnetic multilayer. A slice of the magnetization inside the box, indicated by the dashed line, is shown next to the bounding box. b) When the cocoon emerges to the surface of the magnetic multilayer, one side of the box displays a skyrmion texture, yielding a topological charge of ±1. c) What discretization preserves the 3D topology? As the symmetric exchanged is reduced, keeping only the dipolar coupling between the layers, no more continuity of the magnetization is expected between layers, and the Bloch points can be truly avoided.